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Integration Question

  1. May 7, 2010 #1
    1. The problem statement, all variables and given/known data
    Hi, I know this is a mechanics question, but I don't think the actual problem I have with it involves any mechanics, it's just integration techniques.

    Find the deflection angle of a particle moving in the following repulsive central field:

    U = α/r², α > 0




    2. Relevant equations

    Use the formula [tex]\int[/tex]1/(x√(x² - 1))dx = π/2 (π = pi)
    where the integral limits are 1 (lower) and ∞ (upper)


    3. The attempt at a solution
    Hi everyone, here's what I've done so far:

    I use the formula χ = | π - 2ϕ_0 |, where χ is the angle of deflection

    and then ϕ_0 = [tex]\int[/tex] (ρ/r²√(1 - ρ²/r² - U(r)/E) dr

    where the integral limits are r_min (lower) and ∞ (upper)

    I am trying to turn this into the form given in the question to apply the formula.

    First I factor out a 1/r from inside the square root and sub in the value for U(r):

    ϕ_0 = [tex]\int[/tex] (ρ/r√(r² - (ρ² + α/E)) dr


    But this is where I get stuck, as I can't see how to turn the (ρ² + α/E) into a 1. Can anyone please point me in the right direction?

    Thanks in advance for any help! :)
     
  2. jcsd
  3. May 7, 2010 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi Pyroadept! :smile:

    (have an integral: ∫ :wink:)

    Substitute r = [√(r² - (ρ² + α/E))]s :wink:

    (btw, a simple trig substitution will give you that integral anyway)
     
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